WEBVTT

1
00:00:00.000 --> 00:00:03.878
Here is the problem: d by d x(2 to the power x.

2
00:00:04.328 --> 00:00:08.345
Find the derivative of (2 to the power x with respect to x.

3
00:00:09.145 --> 00:00:16.866
Exponential rule: d by d x[aᵘ] equals aᵘ times ln(a) times u prime.

4
00:00:17.316 --> 00:00:18.930
Combine and simplify.

5
00:00:19.730 --> 00:00:24.989
So the answer is f prime of x equals 2 to the power x times ln(2).

6
00:00:25.439 --> 00:00:38.384
The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

7
00:00:39.184 --> 00:00:45.964
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.